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853:{\displaystyle {\begin{aligned}\displaystyle \mathrm {Nat} (H^{n}(-,\pi ),H^{q}(-,G))&=\mathrm {Nat} (,)\\&=\\&=H^{q}(K(\pi ,n);G).\end{aligned}}}
83:, in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map, at
927:
895:
122:, about which information is hard to come by. This connection established the deep interest of the cohomology operations for
222:
127:
871:
114:
of Hom-functors; if there is a bicommutant aspect, taken over the
Steenrod algebra acting, it is only at a
953:
374:
316:
434:
167:
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119:
41:
428:
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has its own cohomology operations, and these may exhibit a richer set on constraints.
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25:
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922:, Annals of Mathematics Studies, vol. 50, Princeton University Press,
320:
107:
56:
28:, from the 1950s onwards, in the shape of the simple definition that if
300:
63:
The origin of these studies was the work of
Pontryagin, Postnikov, and
91:
of operations has been brought into close relation with that of the
368:
84:
33:
431:
once again, the cohomology operation is given by an element of
887:
Cohomology operations and applications in homotopy theory
52:
the operations can be studied by combinatorial means; and
55:
the effect of the operations is to yield an interesting
48:
to itself. Throughout there have been two basic points:
306:
579:
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289:{\displaystyle \theta :H^{n}(-,\pi )\to H^{q}(-,G)\,}
225:
170:
147:
852:
555:
535:
515:
480:
419:
359:
288:
201:
153:
945:
523:denote the set of homotopy classes of maps from
884:Mosher, Robert E.; Tangora, Martin C. (2008) ,
126:, and has been a research topic ever since. An
883:
40:, then a cohomology operation should be a
285:
198:
911:
118:level. The convergence is to groups in
946:
307:Relation to Eilenberg–MacLane spaces
133:
420:{\displaystyle K(\pi ,n)\to K(G,q)}
13:
665:
662:
659:
587:
584:
581:
481:{\displaystyle H^{q}(K(\pi ,n),G)}
14:
965:
106:aspect is implicit in the use of
87:level. The general theory of the
890:, New York: Dover Publications,
323:a cohomology operation of type
128:extraordinary cohomology theory
872:Secondary cohomology operation
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311:Cohomology of CW complexes is
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202:{\displaystyle (n,q,\pi ,G)\,}
195:
171:
1:
877:
360:{\displaystyle (n,q,\pi ,G)}
7:
865:
10:
970:
20:concept became central to
912:Steenrod, N. E. (1962),
67:, who first defined the
317:Eilenberg–MacLane space
154:{\displaystyle \theta }
100:Adams spectral sequence
854:
557:
537:
517:
491:Symbolically, letting
482:
421:
361:
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214:natural transformation
203:
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120:stable homotopy theory
42:natural transformation
919:Cohomology operations
855:
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547:
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140:cohomology operation
18:cohomology operation
16:In mathematics, the
81:singular cohomology
954:Algebraic topology
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650:
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417:
357:
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22:algebraic topology
929:978-0-691-07924-0
914:Epstein, D. B. A.
897:978-0-486-46664-4
556:{\displaystyle B}
536:{\displaystyle A}
134:Formal definition
69:Pontryagin square
38:cohomology theory
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516:{\displaystyle }
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429:representability
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112:derived functors
89:Steenrod algebra
73:Postnikov square
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124:homotopy theory
93:symmetric group
79:operations for
77:Steenrod square
65:Norman Steenrod
26:homotopy theory
24:, particularly
12:
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5:
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416:
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410:
407:
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401:
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389:
386:
383:
380:
371:class of maps
367:is given by a
356:
353:
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308:
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2:
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888:
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314:
313:representable
304:
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58:
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50:
49:
47:
43:
39:
35:
31:
27:
23:
19:
918:
886:
490:
321:Yoneda lemma
319:, so by the
310:
301:CW complexes
298:
216:of functors
211:
139:
137:
115:
108:Ext functors
103:
97:
62:
45:
29:
17:
15:
299:defined on
104:bicommutant
57:bicommutant
36:defining a
878:References
823:π
758:π
709:−
688:π
676:−
636:−
614:π
608:−
458:π
397:→
385:π
346:π
274:−
258:→
252:π
246:−
227:θ
187:π
161:of type
149:θ
948:Category
866:See also
427:. Using
369:homotopy
938:0145525
916:(ed.),
906:0226634
116:derived
98:In the
85:cochain
59:theory.
34:functor
936:
926:
904:
894:
315:by an
110:, the
75:, and
212:is a
44:from
32:is a
924:ISBN
892:ISBN
102:the
543:to
950::
934:MR
932:,
902:MR
900:,
563:,
488:.
303:.
138:A
95:.
71:,
844:.
841:)
838:G
835:;
832:)
829:n
826:,
820:(
817:K
814:(
809:q
805:H
801:=
791:]
788:)
785:q
782:,
779:G
776:(
773:K
770:,
767:)
764:n
761:,
755:(
752:K
749:[
746:=
736:)
733:]
730:)
727:q
724:,
721:G
718:(
715:K
712:,
706:[
703:,
700:]
697:)
694:n
691:,
685:(
682:K
679:,
673:[
670:(
666:t
663:a
660:N
656:=
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645:)
642:G
639:,
633:(
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624:H
620:,
617:)
611:,
605:(
600:n
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588:t
585:a
582:N
551:B
531:A
511:]
508:B
505:,
502:A
499:[
476:)
473:G
470:,
467:)
464:n
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455:(
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391:n
388:,
382:(
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349:,
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340:q
337:,
334:n
331:(
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280:G
277:,
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230::
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193:G
190:,
184:,
181:q
178:,
175:n
172:(
46:F
30:F
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